Question

Let f (x) = sin(ax + b) and let g (x) = cos(ax + b) where a and b are constants. Gues a formular for f*(x) and g"(x) for general n (positive integer )

Answer 1

Given $f (x) = sin(ax + b), g(x) = cos(ax+b)$

$\implies f'(x) = a \ cos(ax+b), g'(x) = -a \ sin(ax+b)$

$\implies f''(x) = -a^2 sin(ax+b), g''(x)=-a^2cos(ax+b)$

$\implies f'''(x) = -a^3 cos(ax+b), g'''(x) = a^3 sin(ax+b) \\ \implies f^{iv}(x) = a^4 sin(ax+b), g^{iv}(x) = a^4 cos(ax+b)$

Hence In general,

$f^n(x) = \begin{cases} (-1)^{\frac{n}{2}} a^n sin(ax+b) : n \ is \ even \\ (-1)^{\frac{n-1}{2}} a^n cos(ax+b) : n \ is \ odd \end{cases}$

and $g^n(x) = \begin{cases} (-1)^{\frac{n}{2}} a^n cos(ax+b) : n \ is \ even \\ (-1)^{\frac{n+1}{2}} a^n sin(ax+b) : n \ is \ odd \end{cases}$